Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials

Abstract

A. H. Stroud has recently shown the existence of cubature formulas for planar regions which use m 2 {m^2} points and have polynomial precision 2 m − 1 2m - 1 . In this paper, the author gives sufficient conditions for the existence of formulaa using fewer than m 2 {m^2} points, and having polynomial precision 2 m − 1 2m - 1 . An algorithm is given for computing such formulas, and is shown to be useful in a more general setting than given in the theorem. Numerical examples are given, both in terms of previously known and new cubature formulas.